LIMIT-THEOREMS FOR THE SUPERPROCESS WITH IMMIGRATION

In this note we study the limiting behavior of a certain class of superprocess with immigration. Such a process X(t) is non-extinction almost surely due to external particles' immigrating. Under suitable conditions, which include the convergence of the semigroup for the underlying process to some limiting probability measure v, we show that the random measure t-1 X(t) converges in distribution to Z(c)v as t --> infinity, where Z(c) is a random variable possessing GAMMA-distribution with parameter c. Moreover, for the weighted occupation time process Y(t), we prove that t-2 Y(t) converges in distribution to U(c)v, where U(c) is a deterministic random variable.